\documentclass[12pt]{article}\usepackage{graphicx}\usepackage{amssymb}\usepackage{epstopdf}\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}\textwidth 4.5in\hoffset 1in\voffset 2in\textheight = 8in\oddsidemargin = 0.0 in\evensidemargin = 0.0 in\topmargin = 0.2 in\headheight = -2.0 in\headsep = -1.0 in\parskip = 0.2in\parindent = 0.0in\newtheorem{theorem}{Theorem}\newtheorem{corollary}[theorem]{Corollary}\newtheorem{definition}{Definition}\def\today{}

\title{About the Schwarz  H family}
\author{H. Karcher}
\begin{document}
\maketitle



  This is a 1-parameter family of triply periodic surfaces. The
surfaces are made of pieces which one could call ``triangular
catenoids''; annular Plateau solutions bounded by two parallel
equilateral triangles. In the morphing  aa  changes the
height-to-edge length ratio of these triangular catenoids. Observe
that, as in the case of circles bounding catenoids, the distance
between the triangles has to be small enough and then they
bound a stable and an unstable triangular catenoid. In the
PD-family with dd=0 one can observe analogous ``square
catenoids'', except that our parametrization does not emphasize
them.

   When Alan Schoen found more triply periodic surfaces around
1970 he named these ``Schwarz' H surfaces''. (Maybe Schwarz
constructed only one member of the family.)  Later, the Swedish 
chemist, Lidin,  found another embedded example (now called 
the Lidinoid) in the same associate
family, when aa is approximately 0.55, and the associate family 
parameter is 0.7139, (about 64.25 degrees).


  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991

 \end{document}